The idea, step by step
Four conditions must work together
A binomial variable X counts successes in a fixed number n of trials. Each trial must have two success/failure categories, a common success probability p, and outcomes independent across trials. Write only after identifying X, n and p.
Different underlying outcomes may be grouped into success/failure categories. A die roll has six scores, but “roll at least 5” versus “do not” gives two categories. The total number of such successes in a fixed set of independent rolls can be binomial.
Look closely at the process
Sampling without replacement from a small group changes the chance of success after each draw, so it is not exactly binomial. Repeated customer outcomes may be dependent because of shared weather or special events. A campaign that changes its success rate partway through violates constant p.
A large population with a very small sampling fraction may justify an approximate binomial model, but the approximation must be stated and defended. Do not treat a plausible context as proof that all conditions hold. A process stopping at the first success does not have fixed n.
A common trap
Any number of successes must be binomial.
Instead: A count needs the fixed-trial, two-category, independence and common-probability conditions. The word count alone establishes none of them.
See the thinking
An appropriate grouping
A fair die is rolled independently 12 times. X counts scores of 5 or 6. State a model and justify it.
- There are 12 fixed trials. Define success as a score of 5 or 6.
- The success probability on each trial is 2/6 = 1/3.
- The rolls are independent and success/failure categories exhaust the scores.
Why replacement matters
A bag has two red and three blue counters. Two are drawn without replacement. Explain why the red count is not exactly Bin(2, 0.4).
- The first red probability is 2/5.
- After a red draw it becomes 1/4; after a blue draw it becomes 2/4.
- The outcomes are dependent and the conditional probabilities change.
Check your understanding
A fair coin is tossed until its first head. Is the number of tails before stopping binomial?
Show answer
No.
The stopping rule makes the number of trials variable rather than fixed.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- A fair die is rolled independently 8 times. X counts even scores. State n and p.2 marks
- Ten bulbs are tested independently, each with fault probability 0.03. State a model for the number of faults.2 marks
- Four cards are drawn without replacement from a standard deck. Explain why the heart count is not exactly Bin(4, 1/4).2 marks
- A player takes 20 independent shots, with success probability 0.8 for the first 10 and 0.6 for the last 10. Why is the total not exactly Bin(20, 0.7)?2 marks
- An online poll stops when it has collected five positive responses. Identify the fixed-n problem.2 marks
- A factory samples 10 items without replacement from 100,000 items. Explain a possible approximate binomial model and a remaining limitation.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A fair die is rolled independently 8 times. X counts even scores. State n and p.2 marks
There are three even scores out of six.
The stated independent, fixed rolls supply the other binomial conditions.
. - Ten bulbs are tested independently, each with fault probability 0.03. State a model for the number of faults.2 marks
Success means a fault.
There are ten fixed independent tests with common success probability 0.03.
. - Four cards are drawn without replacement from a standard deck. Explain why the heart count is not exactly Bin(4, 1/4).2 marks
After drawing a heart there are 12 hearts among 51 cards.
After drawing a non-heart there are 13 hearts among 51; neither equals the unconditional first-draw probability.
The draws are dependent and the heart probability changes. - A player takes 20 independent shots, with success probability 0.8 for the first 10 and 0.6 for the last 10. Why is the total not exactly Bin(20, 0.7)?2 marks
The mean probability being 0.7 does not make every trial have probability 0.7.
The total is a sum of two counts with different success probabilities.
The success probability is not constant across trials. - An online poll stops when it has collected five positive responses. Identify the fixed-n problem.2 marks
The stopping condition fixes successes, not trials.
A binomial model requires a predetermined number of trials.
The number of people polled is random. - A factory samples 10 items without replacement from 100,000 items. Explain a possible approximate binomial model and a remaining limitation.3 marks
State n = 10 and estimate p from appropriate process information.
The draws are still not exactly independent, and clustering or a changing production process could undermine the approximation.
The sample fraction is tiny, so the fault probability changes little; independence is approximate.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 Mathematical Methods: discrete random variables and probability distributions, expected value and variance, transformations and independent sums, Bernoulli and binomial models. Original supporting lessons and exercises follow the approved Ascent chapter structure.
Official curriculum source ↗